Track / Stage 3 / 116 of 260 #116 of 1964

Problem 116

AMC 10/12, early questions

Number theory Difficulty 3.5 Multiple choice

Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin $p_0=(0,0)$ facing to the east and walks one unit, arriving at $p_1=(1,0)$ . For $n=1,2,3,\dots$ , right after arriving at the point $p_n$ , if Aaron can turn $90^\circ$ left and walk one unit to an unvisited point $p_{n+1}$ , he does that. Otherwise, he walks one unit straight ahead to reach $p_{n+1}$ . Thus the sequence of points continues $p_2=(1,1), p_3=(0,1), p_4=(-1,1), p_5=(-1,0)$ , and so on in a counterclockwise spiral pattern. What is $p_{2015}$ ?

Pick one

Official solution

The first thing we would do is track Aaron's footsteps:
He starts by taking $1$ step East and $1$ step North, ending at $(1,1)$ after $2$ steps and about to head West.
Then he takes $2$ steps West and $2$ steps South, ending at $(-1,-1$ ) after $2+4$ steps, and about to head East.
Then he takes $3$ steps East and $3$ steps North, ending at $(2,2)$ after $2+4+6$ steps, and about to head West.
Then he takes $4$ steps West and $4$ steps South, ending at $(-2,-2)$ after $2+4+6+8$ steps, and about to head East.
From this pattern, we can notice that for any integer $k \ge 1$ he's at $(-k, -k)$ after $2 + 4 + 6 + ... + 4k$ steps, and about to head East. There are $2k$ terms in the sum, with an average value of $(2 + 4k)/2 = 2k + 1$ , so:
$2 + 4 + 6 + ... + 4k = 2k(2k + 1)$
If we substitute $k = 22$ into the equation: $44(45) = 1980 < 2015$ . So he has $35$ moves to go. This makes him end up at $(-22+35,-22) = (13,-22) \implies \boxed{\textbf{(D)} (13, -22)}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.